1. Prolog Line, surface & volume integrals in n-D space → Exterior forms Time evolution of such integrals → Lie derivatives Dynamics with constraints → Frobenius theorem on differential forms Curvatures → Differential geometry –Spacetime curvatures ~ General relativity –Field space curvatures ~ Gauge theories Symmetries of quantum fields → Lie groups Existence & uniqueness of problem → Topology –Examples: Homology groups, Brouwer degree, Hurewicz homotopy groups, Morse theory, Atiyah-Singer index theorem, Gauss-Bonnet- Poincare theorem, Chern characteristic classes Website:http://ckw.phys.ncku.edu.twhttp://ckw.phys.ncku.edu.tw Homework submission:class@ckw.phys.ncku.edu.tw

2. The Geometry of Physics, An Introduction, 2 nd ed. T. Frankel Cambridge University Press (97, 04) I. Manifolds, Tensors, & Exterior Forms II. Geometry & Topology III. Lie Groups, Bundles, & Chern Forms

3. I. Manifolds, Tensors, & Exterior Forms 1.Manifolds & Vector Fields 2.Tensors, & Exterior Forms 3.Integration of Differential Forms 4.The Lie Derivative 5.The Poincare Lemma & Potentials 6.Holonomic & Nonholonomic Constraints

4. II. Geometry & Topology 7.R 3 and Minkowski Space 8.The Geometry of Surfaces in R 3 9.Covariant Differentiation & Curvature 10.Geodesics 11.Relativity, Tensors, & Curvature 12.Curvature & Simple Connectivity 13.Betti Numbers & De Rham's Theorem 14.Harmonic Forms

5. III. Lie Groups, Bundles, & Chern Forms 15.Lie Groups 16.Vector Bundles in Geometry & Physics 17.Fibre Bundles, Gauss-Bonnet, & Topological Quantization 18.Connections & Associated Bundles 19.The Dirac Equation 20.Yang-Mills Fields 21.Betti Numbers & Covering Spaces 22.Chern Forms & Homotopy Groups

6. Supplementary Readings Companion textbook: –C.Nash, S.Sen, "Topology & Geometry for Physicists", Acad Press (83) Differential geometry (standard references) : –M.A.Spivak, "A Comprehensive Introduction to Differential Geometry" ( 5 vols), Publish or Perish Press (79) –S.Kobayashi, K.Nomizu, "Foundations of Differential Geometry" (2 vols), Wiley (63) Particle physics: –A.Sudbery, "Quantum Mechanics & the Particles of Nature", Cambridge (86)

7. 1. Manifolds & Vector Fields 1.1. Submanifolds of Euclidean Space 1.2. Manifolds 1.3. Tangent Vectors & Mappings 1.4. Vector Fields & Flows

8. 1.1. Submanifolds of Euclidean Space 1.1.a. Submanifolds of R N. 1.1.b. The Geometry of Jacobian Matrices: The "differential". 1.1.c. The Main Theorem on Submanifolds of R N. 1.1.d. A Non-trivial Example: The Configuration Space of a Rigid Body

9. 1.1.a. Submanifolds of R N A subset M = M n  R n+r is an n-D submanifold of R n+r if  P  M,  a neiborhood U in which P can be described by some coordinate system of R n+r  whereare differentiable functions are called local (curvilinear) coordinates in U